How does DigitalOcean and Discourse work?

I have a basic question on how the droplets work. Lets say i have a website www.builderstatus.com which is live but i want to have a forum for it at say forum.builderstatus.com. Can i have this subdomain will be serviced by digital ocean droplet while rest of the site will be from hostgator (where i have hosted). Is this possible? Is this possible or the whole site needs to be hosted in digital ocean?

What you'd want to do is deploy a Droplet and get the IP address from your control panel. Once you have the Droplets' Public IP, you need to create a DNS entry for your sub-domain and point it to that IP.

For this specific case, you'd want to create an A entry.

Of course, you'll need to either 1). Deploy a one-click Droplet; or 2). Configure the VPS on your own.

Keep in mind, Droplets (VPS's) with DigitalOcean do not provide cPanel or a similar CP (such as what HostGator offers), so you either need to deploy a one-click Droplet or manually configure the Droplet by installing Apache/NGINX, MySQL/MariaDB, PHP, etc. Droplets are barebones and come with root access only (unless you do the one-click deployments).